Stateline

Stateline Shipping and Transport Company

MAT540: Quantitative Methods

9 June 2011

1. Transportation Model for shipping waste directly from the 6 plants to the 3 waste disposal sites.

The following table gives the cost of shipping per barrel of waste, from each of the six plants to each of the three waste disposal sites, together with amount of waste available in the various plants and capacity of the disposal sites.

  | White water | Los Canos | Duras | Availability |

Kingsport | $12.00 | $15.00 | $17.00 | 35 |

Danville | $14.00 | $9.00 | $10.00 | 26 |

Macon | $13.00 | $20.00 | $11.00 | 42 |

Selma | $17.00 | $16.00 | $19.00 | 53 |

Columbus | $7.00 | $14.00 | $12.00 | 29 |

Allentown | $22.00 | $16.00 | $18.00 | 38 |

Capacity | 65 | 80 | 105 | 223 |

The objective of the problem is to develop a shipping schedule that minimizes the total cost of transportation.

Suppose Xij denotes the number of barrels of wastes to be transported from the ith plant to jth site. Then the total cost of transportation is:

Z = 12 X11 + 15 X12 + 17 X13 + 14 X21 + 9 X22 + 10 X23 + 13 X31 + 20 X32 + 11 X33 + 17 X41+ 16 X42 + 19 X43 + 7 X51 + 14 X52+ 12 X53 + 22 X61 + 16 X62 + 18 X63.

Thus the objective function of the problem is to minimize

Z = 12 X11 + 15 X12 + 17 X13 + 14 X21 + 9 X22 + 10 X23 + 13 X31 + 20 X32 + 11 X33 + 17 X41+ 16 X42 + 19 X43 + 7 X51 + 14 X52+ 12 X53 + 22 X61 + 16 X62 + 18 X63

Constraints

Availability in plants:

X11 + X12 + X13 = 35

X21 + X22 + X23 = 26

X31 + X32 + X33 = 42

X41 + X42 + X43 = 53

X51 + X52 + X53 = 29

X61 + X62 + X63 = 38

Capacity of the sites:

X11 + X21+ X31+X41 + X51 + X61 ≤ 65

X12 + X22+ X32+X42 + X52 + X62 ≤ 80

X13 + X23+ X33+X43 + X53 + X63 ≤ 105

Non- Negativity restrictions

Xij ≥ 0 , i = 1,2,3,4,5,6 ; j = 1,2,3.

The model is:

Minimize:

Z = 12 X11 + 15 X12 + 17 X13 + 14 X21 + 9 X22 + 10 X23 + 13 X31 + 20 X32 + 11 X33 + 17 X41+ 16 X42 + 19 X43 + 7 X51 + 14 X52+ 12 X53 + 22 X61...